Constraints

OLAF uses grid units for the locational decision variables and meter as the physical dimension of the stack height resp. diameter. It is therefore comparatively simple to add constraints to the optimization process. The following constraints with respect to the decision variables have been added:

1. Locational constraints:
   1. The facility has to lie inside a circle with center at W longitude and N latitude and radius 180.738km (20 grid units). This translated readily into the convex quadratic constraint
      
      
   2. For hypothetical political reasons, the facility is not allowed to lie within a circle of radius 100km around Paris and within circles of radius 60km around Brussels and Reims. With a grid spacing of 9.0869km, these constraints can be expressed as concave quadratic constraints:
      
      for Brussels,
      
      for Paris, and
      
      for Reims (cmp. Figure 6.9).

       
      Figure 6.9: The feasible locational region (white area).  
      

2. Engineering constraints:
   1. The stack height has to be between 20m and 90m, i. e.\
      
      
   2. The stack diameter at the top has to be between 2 m and 7m, i. e.\
      
      
   Note that these simple box constraints for the physical characteristics of the stack are rather unrealistic. More realistic constraints have to include a nonlinear coupling between h and w.